<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V4">complete</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="conlat_1.html#NM1">FormalContext</a>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:22_1"/>
 <a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font>,<font color="Maroon">L</font> <a href="filter_1.html#R1">are_isomorphic</a> 
 ;<br/><b>then consider </b><font color="Maroon">f</font> being    <a href="lattice4.html#M1">Homomorphism</a> of  <a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:22_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font> is <a href="lattice4.html#V3">isomorphism</a>
 <b>by </b><i><a class="ref" href="lattice4.html#D5">LATTICE4:def 5</a></i>;<br/><a NAME="E4:22_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">f</font> is <a href="lattice4.html#V1">monomorphism</a> &amp; <font color="Maroon">f</font> is <a href="lattice4.html#V2">epimorphism</a> )
 <b>by </b><i>, <a class="ref" href="lattice4.html#D4">LATTICE4:def 4</a></i>;<br/><b>take </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span>;<br/><b>take </b><font color="Maroon">d</font> = <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span>;<br/><b>thus </b><a NAME="E5:22_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> is <a href="lattice6.html#V8">supremum-dense</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice6.html#D13">LATTICE6:def 13</a><br/>

<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span><b> such that </b><br/><a NAME="E2:22_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="lattice4.html#T9">LATTICE4:9</a></i>;<br/>
<a NAME="E3:22_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span> is <a href="lattice6.html#V8">supremum-dense</a>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">D'</font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:22_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">D'</font>,<span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice6.html#D13">LATTICE6:def 13</a></i>;<br/>
<a NAME="E6:22_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">D'</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>  st <font color="Maroon">D'</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Olive">r</font> holds <br/><font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Olive">r</font> ) )
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<b>set </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span> ;<br/>
<a NAME="E7:22_1_1"/><i><font color="Green">E40</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span><b> such that </b><br/><a NAME="E3:22_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> )
 ;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:22_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:22_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="txt" href="conlat_2.html#E3:22_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E7:22_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:22_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<b>hence </b><a NAME="E9:22_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:22_1_1"/><i><font color="Green">E46</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span>  <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:22_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">q'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span><b> such that </b><br/><a NAME="E3:22_1_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> )
 ;<br/>
<a NAME="E4:22_1_1_2"/>
<font color="Maroon">q'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="txt" href="conlat_2.html#E3:22_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>
<b>hence </b><a NAME="E5:22_1_1_2"/>
<font color="Maroon">q</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="txt" href="conlat_2.html#E3:22_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="lattice4.html#T8">LATTICE4:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:22_1_1"/>
for <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span>  <a href="lattice3.html#R4">is_less_than</a> <font color="Olive">r</font> holds <br/><font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:22_1_1_3"/><i><font color="Green">E49</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span>  <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">r</font>
 ;<br/>

<b>consider </b><font color="Maroon">r'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span><b> such that </b><br/><a NAME="E3:22_1_1_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r'</font>
 <b>by </b><i>, <a class="ref" href="lattice4.html#T9">LATTICE4:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r'</font> = <font color="Maroon">r'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> ;<br/>
<a NAME="E5:22_1_1_3"/>
for <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> holds <br/><font color="Olive">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:22_1_1_3_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font>
 ;<br/>

<a NAME="E2:22_1_1_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span> 
 
;<br/>
<a NAME="E3:22_1_1_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r'</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>
<b>hence </b><a NAME="E4:22_1_1_3_1"/>
<font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r'</font>
 <b>by </b><i>, <a class="ref" href="lattice4.html#T8">LATTICE4:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_1_1_3"/><b>then </b>
<font color="Maroon">D'</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">r'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>
<a NAME="E7:22_1_1_3"/><b>then </b>
<font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r'</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E8:22_1_1_3"/>
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="lattice4.html#T8">LATTICE4:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:22_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E11:22_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E6:22_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">d</font> is <a href="lattice6.html#V9">infimum-dense</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice6.html#D14">LATTICE6:def 14</a><br/>

<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span><b> such that </b><br/><a NAME="E2:22_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="lattice4.html#T9">LATTICE4:9</a></i>;<br/>
<a NAME="E3:22_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> is <a href="lattice6.html#V9">infimum-dense</a>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">D'</font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:22_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">D'</font>,<span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice6.html#D14">LATTICE6:def 14</a></i>;<br/>
<a NAME="E6:22_1_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">b</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">D'</font> &amp; ( for <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>  st <font color="Olive">r</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">D'</font> holds <br/><font color="Olive">r</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> ) )
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<b>set </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span> ;<br/>
<a NAME="E7:22_1_2"/><i><font color="Green">E54</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span><b> such that </b><br/><a NAME="E3:22_1_2_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> )
 ;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:22_1_2_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:22_1_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="txt" href="conlat_2.html#E3:22_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E7:22_1_2_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:22_1_2_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<b>hence </b><a NAME="E9:22_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:22_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">a</font> <a href="lattice3.html#R3">is_less_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:22_1_2_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">q'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span><b> such that </b><br/><a NAME="E3:22_1_2_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> )
 ;<br/>
<a NAME="E4:22_1_2_2"/>
<font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q'</font>
 
<b>by </b><i>, <a class="txt" href="conlat_2.html#E3:22_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<b>hence </b><a NAME="E5:22_1_2_2"/>
<font color="Maroon">a</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">q</font>
 <b>by </b><i>, , <a class="txt" href="conlat_2.html#E3:22_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="lattice4.html#T8">LATTICE4:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:22_1_2"/>
for <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">r</font> <a href="lattice3.html#R3">is_less_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span>  holds <br/><font color="Olive">r</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:22_1_2_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">r</font> <a href="lattice3.html#R3">is_less_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">r'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span><b> such that </b><br/><a NAME="E3:22_1_2_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r'</font>
 <b>by </b><i>, <a class="ref" href="lattice4.html#T9">LATTICE4:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r'</font> = <font color="Maroon">r'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> ;<br/>
<a NAME="E5:22_1_2_3"/>
<font color="Maroon">r'</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">D'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:22_1_2_3_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font>
 ;<br/>

<a NAME="E2:22_1_2_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span> 
 
;<br/>
<a NAME="E3:22_1_2_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<b>hence </b><a NAME="E4:22_1_2_3_1"/>
<font color="Maroon">r'</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="lattice4.html#T8">LATTICE4:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_1_2_3"/><b>then </b>
<font color="Maroon">r'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<b>hence </b><a NAME="E7:22_1_2_3"/>
<font color="Maroon">r</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="lattice4.html#T8">LATTICE4:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:22_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<b>hence </b><a NAME="E11:22_1_2"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">d</font></span>)</span> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">o</font> be   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>;<br/><a NAME="E7:22_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">a</font> iff <span class="p1">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:22_1_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:22_1_3"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:22_1_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E4:22_1_3"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , , <a class="ref" href="lattice4.html#T8">LATTICE4:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E9:22_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E10:22_1"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:22_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E12:22_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="lattice4.html#T8">LATTICE4:8</a></i>;<br/><b>hence </b><a NAME="E13:22_1"/>
<font color="Maroon">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>, <font color="Maroon">d</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><a NAME="E2:22"/><i><font color="Green">E118</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> is <a href="lattice6.html#V8">supremum-dense</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">d</font> is <a href="lattice6.html#V9">infimum-dense</a> &amp; ( for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">a</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> holds <br/> ( <font color="Olive">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a</font> iff <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> ) ) )
 ;<br/>

<b>set </b><font color="Maroon">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #),<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">O</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>, <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : (  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Olive">O</font> </span>}</span> ,<font color="Maroon">L</font> ) </span>}</span> ;<br/>
<a NAME="E3:22"/><i><font color="Green">E119</font></i>: 
 <a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="conlat_1.html#K8">B-carrier</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K10">B-join</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K9">B-meet</a> <font color="Maroon">C</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="conlat_1.html#D23">CONLAT_1:def 23</a></i>;<br/>
<a NAME="E4:22"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #),<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">O</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>, <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : (  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Olive">O</font> </span>}</span> ,<font color="Maroon">L</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #),<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">O</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>, <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : (  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Olive">O</font> </span>}</span> ,<font color="Maroon">L</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">O</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>, <font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:22_2"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #),<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> </span>}</span> ,<font color="Maroon">L</font> )
 ;<br/>
<a NAME="E4:22_2"/>
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="conlat_1.html#T35">CONLAT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E5:22_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #),<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">O</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>, <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : (  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Olive">O</font> </span>}</span> ,<font color="Maroon">L</font> ) </span>}</span>  as    <a href="relset_1.html#M2">Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E6:22"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>

<a NAME="E2:22_3"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">y</font> is  <a href="conlat_1.html#V6">strict</a> <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="conlat_1.html#T35">CONLAT_1:35</a></i>;<br/>
<b>then consider </b><font color="Maroon">O'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:22_3"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #)
 ;<br/>
<b>reconsider </b><font color="Maroon">z</font> =  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font> </span>}</span> ,<font color="Maroon">L</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E6:22_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:22_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22"/><b>then </b><i><font color="Green">E124</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E8:22"/>
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_4"/><i><font color="Green">E127</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">O</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>, <font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:22_4"/><i><font color="Green">E128</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #),<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> </span>}</span> ,<font color="Maroon">L</font> )
 ;<br/>
<b>consider </b><font color="Maroon">O'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>, <font color="Maroon">z'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:22_4"/><i><font color="Green">E130</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #),<font color="Maroon">z'</font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font> </span>}</span> ,<font color="Maroon">L</font> )
 <b>by </b><i/>;<br/>
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #)
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<b>hence </b><font color="Maroon">y1</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">y2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="funct_1.html#D1">FUNCT_1:def 1</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:22"/><i><font color="Green">E131</font></i>: 
 <a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="conlat_1.html#K8">B-carrier</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K10">B-join</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K9">B-meet</a> <font color="Maroon">C</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="conlat_1.html#D23">CONLAT_1:def 23</a></i>;<br/>
<a NAME="E11:22"/><i><font color="Green">E132</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>  st <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">b</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:22_5"/>
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:22_5"/><b>then </b>
<font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a>  <a href="conlat_1.html#R2">is-SubConcept-of</a> <font color="Maroon">b</font> <a href="conlat_1.html#K12">@</a> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#T47">CONLAT_1:47</a></i>;<br/>
<a NAME="E3:22_5"/><b>then </b><i><font color="Green">E133</font></i>: 
the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D19">CONLAT_1:def 19</a></i>;<br/>
<a NAME="E4:22_5"/><i><font color="Green">E134</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">o</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_5_1"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> )
 ;<br/>
<b>thus </b><a NAME="E4:22_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">xa</font> =  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font>, <font color="Maroon">xb</font> =  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E6:22_5"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span>,<font color="Maroon">xa</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="conlat_1.html#K12">@</a> </span>)</span>,<font color="Maroon">xb</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> )
 
;<br/>
<a NAME="E7:22_5"/><b>then </b><i><font color="Green">E138</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">xa</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">xb</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E8:22_5"/>
( <font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="conlat_1.html#K12">@</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>;<br/>
<b>hence </b><a NAME="E9:22_5"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#T46">LATTICE3:46</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">fi</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">O</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : ( <font color="Olive">O</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Olive">x</font> </span>}</span>  &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span>  ) </span>}</span> ;<br/>
<a NAME="E12:22"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">O</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : ( <font color="Olive">O</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Olive">x</font> </span>}</span>  &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span>  ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">O</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : ( <font color="Olive">O</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Olive">x</font> </span>}</span>  &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span>  ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">O</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_6"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span>  &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span>  )
 ;<br/>
<a NAME="E4:22_6"/><i><font color="Green">E141</font></i>: 
<span class="p1">(<span class="default"><a href="conlat_1.html#K1">ObjectDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U5">Intent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:22_6_1"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K1">ObjectDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U5">Intent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_6_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1">ObjectDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
 ;<br/>

<a NAME="E2:22_6_1_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">a</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) holds <br/><font color="Olive">a</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">o</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D6">CONLAT_1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">u'</font> being   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:22_6_1_1"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; ( for <font color="Olive">a</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) holds <br/><font color="Olive">a</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">u'</font> ) )
 ;<br/>
<a NAME="E5:22_6_1_1"/><i><font color="Green">E144</font></i>: 
for <font color="Olive">v</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span>  holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:22_6_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span> 
 ;<br/>

<a NAME="E2:22_6_1_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">u'</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:22_6_1_1_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_6_1_1"/><i><font color="Green">E146</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span>  <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:22_6_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_6_1_1_2"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> )
 ;<br/>
<a NAME="E4:22_6_1_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:22_6_1_1_2"/>
<font color="Maroon">q</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_6_1_1"/>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span> ,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_1_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">D</font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span><b> such that </b><br/><a NAME="E2:22_6_1_1_3"/><i><font color="Green">E148</font></i>: 
 <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">D</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="lattice6.html#D13">LATTICE6:def 13</a></i>;<br/>
<a NAME="E3:22_6_1_1_3"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">D</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E4:22_6_1_1_3"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_6_1_1_3_1"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_6_1_1_3_1"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:22_6_1_1_3_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>
<b>hence </b><a NAME="E6:22_6_1_1_3_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_6_1_1_3"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">x</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T46">LATTICE3:46</a></i>;<br/>
<a NAME="E6:22_6_1_1_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span>  <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:22_6_1_1_3_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_6_1_1_3_2"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_6_1_1_3_2"/>
<font color="Maroon">u</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_6_1_1_3"/><b>then </b>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E8:22_6_1_1_3"/>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span> ,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:22_6_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E9:22_6_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:22_6_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
 ;<br/>

<b>then consider </b><font color="Maroon">u'</font> being   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:22_6_1"/><i><font color="Green">E154</font></i>: 
( <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:22_6_1"/><i><font color="Green">E155</font></i>: 
for <font color="Olive">v</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span>  holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:22_6_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">v'</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_6_1_2"/><i><font color="Green">E157</font></i>: 
( <font color="Maroon">v'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_6_1_2"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font>
 <b>by </b><i>, , <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_6_1"/>
for <font color="Olive">v</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span>  holds <br/><font color="Olive">v</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">u'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:22_6_1_3"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> </span>}</span> 
 ;<br/>

<a NAME="E2:22_6_1_3"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:22_6_1_3"/>
<font color="Maroon">v</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">u'</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_6_1"/><b>then </b>
<font color="Maroon">u'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">a</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) holds <br/><font color="Olive">a</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">o</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:22_6_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1">ObjectDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
 <b>by </b><i>, <a class="ref" href="conlat_1.html#D6">CONLAT_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_6"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U5">Intent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:22_6_2"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U5">Intent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_6_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U5">Intent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
 ;<br/>

<a NAME="E2:22_6_2_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D7">CONLAT_1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">u'</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:22_6_2_1"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; ( for <font color="Olive">a</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) holds <br/><font color="Maroon">u'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a</font> ) )
 ;<br/>
<a NAME="E5:22_6_2_1"/><i><font color="Green">E159</font></i>: 
for <font color="Olive">v</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span>  holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:22_6_2_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span> 
 ;<br/>

<a NAME="E2:22_6_2_1_1"/><b>then </b>
<font color="Maroon">u'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">v</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:22_6_2_1_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_6_2_1"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="lattice3.html#R3">is_less_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:22_6_2_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_6_2_1_2"/><i><font color="Green">E161</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 ;<br/>
<a NAME="E4:22_6_2_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:22_6_2_1_2"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_6_2_1"/>
 <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> ,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2_1_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">D</font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">d</font></span>)</span><b> such that </b><br/><a NAME="E2:22_6_2_1_3"/><i><font color="Green">E162</font></i>: 
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">D</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="lattice6.html#D14">LATTICE6:def 14</a></i>;<br/>
<a NAME="E3:22_6_2_1_3"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<a NAME="E4:22_6_2_1_3"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_6_2_1_3_1"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_6_2_1_3_1"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">d</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:22_6_2_1_3_1"/>
<font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 
<b>by </b><i>, , <a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<b>hence </b><a NAME="E6:22_6_2_1_3_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_6_2_1_3"/><b>then </b><i><font color="Green">E166</font></i>: 
 <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T46">LATTICE3:46</a></i>;<br/>
<a NAME="E6:22_6_2_1_3"/>
<font color="Maroon">x</font> <a href="lattice3.html#R3">is_less_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:22_6_2_1_3_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_6_2_1_3_2"/><i><font color="Green">E167</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> &amp; <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_6_2_1_3_2"/>
<font color="Maroon">x</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_6_2_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T40">LATTICE3:40</a></i>;<br/>
<b>hence </b><a NAME="E8:22_6_2_1_3"/>
 <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> ,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:22_6_2_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<b>hence </b><a NAME="E9:22_6_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:22_6_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
 ;<br/>

<b>then consider </b><font color="Maroon">u'</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:22_6_2"/><i><font color="Green">E168</font></i>: 
( <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:22_6_2"/><i><font color="Green">E169</font></i>: 
for <font color="Olive">v</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span>  holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:22_6_2_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">v'</font> being   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_6_2_2"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">v'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v'</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_6_2_2"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, , <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_6_2"/>
for <font color="Olive">v</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span>  holds <br/><font color="Maroon">u'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:22_6_2_3"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span> 
 ;<br/>

<a NAME="E2:22_6_2_3"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:22_6_2_3"/>
<font color="Maroon">u'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_6_2"/><b>then </b>
<font color="Maroon">u'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:22_6_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U5">Intent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
 <b>by </b><i>, <a class="ref" href="conlat_1.html#D7">CONLAT_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_6"/><b>then </b>
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font>
 
<b>by </b><i>, , <a class="ref" href="conlat_1.html#D13">CONLAT_1:def 13</a></i>;<br/>
<a NAME="E7:22_6"/><b>then </b>
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="conlat_1.html#T35">CONLAT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E8:22_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">fi</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #)</span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">O</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : ( <font color="Olive">O</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Olive">x</font> </span>}</span>  &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span>  ) </span>}</span>  as    <a href="relset_1.html#M2">Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E14:22"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">fi</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>set </b><font color="Maroon">O</font> = <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span> ;<br/>
<a NAME="E3:22_7"/>
<span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_7_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">u'</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_7_1"/><i><font color="Green">E171</font></i>: 
( <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_7_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">O</font> = <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font> ;<br/>
<a NAME="E5:22_7"/>
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_7_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">u'</font> being   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_7_2"/><i><font color="Green">E172</font></i>: 
( <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_7_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> ;<br/>
<a NAME="E7:22_7"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fi</font>
 
;<br/>
<b>hence </b><a NAME="E8:22_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">fi</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:22"/><b>then </b><i><font color="Green">E173</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">fi</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E16:22"/>
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fi</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fi</font> holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_8"/><i><font color="Green">E174</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fi</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fi</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">O</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_8"/><i><font color="Green">E175</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z</font> </span>}</span>  &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">z</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span>  )
 ;<br/>
<b>consider </b><font color="Maroon">z'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">O'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E5:22_8"/><i><font color="Green">E176</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z'</font>,<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #)</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">O'</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z'</font> </span>}</span>  &amp; <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">z'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> </span>}</span>  )
 <b>by </b><i/>;<br/>
<font color="Maroon">z</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">z'</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<b>hence </b><font color="Maroon">y1</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">y2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">fi</font> = <font color="Maroon">fi</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <b>by </b><i>, <a class="ref" href="funct_1.html#D1">FUNCT_1:def 1</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:22"/><i><font color="Green">E177</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">b</font> holds <br/><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:22_9"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:22_9"/><i><font color="Green">E179</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x'</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_9_1"/><i><font color="Green">E180</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> )
 ;<br/>
<a NAME="E4:22_9_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>
<b>hence </b><a NAME="E5:22_9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:22_9"/><i><font color="Green">E181</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">ya</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:22_9"/><i><font color="Green">E183</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">ya</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fi</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">xa</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">Oa</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">Aa</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E7:22_9"/><i><font color="Green">E186</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">ya</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xa</font>,<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Oa</font>,<font color="Maroon">Aa</font> #)</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">Oa</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">xa</font> </span>}</span>  &amp; <font color="Maroon">Aa</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a'</font> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">xa</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span>  )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">yb</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:22_9"/><i><font color="Green">E188</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">yb</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fi</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">xb</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">Ob</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">Ab</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E11:22_9"/><i><font color="Green">E191</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">yb</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xb</font>,<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ob</font>,<font color="Maroon">Ab</font> #)</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">Ob</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">xb</font> </span>}</span>  &amp; <font color="Maroon">Ab</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a'</font> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">xb</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span>  )
 <b>by </b><i/>;<br/>
<i><font color="Green">E192</font></i>: <font color="Maroon">a</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xa</font>,<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Oa</font>,<font color="Maroon">Aa</font> #)</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">xa</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<i><font color="Green">E193</font></i>: <font color="Maroon">b</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xb</font>,<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ob</font>,<font color="Maroon">Ab</font> #)</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">xb</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<a NAME="E14:22_9"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Oa</font>,<font color="Maroon">Aa</font> #)
 
<b>by </b><i>, , <a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E15:22_9"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ob</font>,<font color="Maroon">Ab</font> #)
 
<b>by </b><i>, <a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>set </b><font color="Maroon">ca</font> = <font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">cb</font> = <font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>;<br/>
<a NAME="E16:22_9"/>
( the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Oa</font> &amp; the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ob</font> )
 
<b>by </b><i>, , <a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>;<br/>
<a NAME="E17:22_9"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="conlat_1.html#K12">@</a>  <a href="conlat_1.html#R2">is-SubConcept-of</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="conlat_1.html#K12">@</a> 
 
<b>by </b><i>, , <a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="conlat_1.html#D19">CONLAT_1:def 19</a></i>;<br/>
<b>hence </b><a NAME="E18:22_9"/>
<font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="conlat_1.html#T47">CONLAT_1:47</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:22"/><i><font color="Green">E198</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>set </b><font color="Maroon">O</font> = <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> </span>}</span> ;<br/>
<a NAME="E2:22_10"/>
<span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_10_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">u'</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_10_1"/><i><font color="Green">E199</font></i>: 
( <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_10_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">O</font> = <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font> ;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">a'</font> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> ;<br/>
<a NAME="E4:22_10"/>
<span class="p1">{<span class="default"> <font color="Olive">a'</font> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_10_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a'</font> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">u'</font> being   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_10_2"/><i><font color="Green">E200</font></i>: 
( <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_10_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">a'</font> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> ;<br/>
<a NAME="E6:22_10"/><i><font color="Green">E201</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fi</font>
 
;<br/>
<b>set </b><font color="Maroon">b</font> =  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) </span>}</span> ,<font color="Maroon">L</font>;<br/>
<a NAME="E7:22_10"/>
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fi</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<a NAME="E8:22_10"/><b>then </b>
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="conlat_1.html#T35">CONLAT_1:35</a></i>;<br/>
<a NAME="E9:22_10"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #),<span class="p2">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) </span>}</span> ,<font color="Maroon">L</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 
;<br/>
<a NAME="E10:22_10"/><b>then </b><i><font color="Green">E202</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E11:22_10"/>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o'</font> where <font color="Olive">o'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> </span>}</span>  </span>}</span> ,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">D</font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span><b> such that </b><br/><a NAME="E2:22_10_3"/><i><font color="Green">E203</font></i>: 
 <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">D</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="lattice6.html#D13">LATTICE6:def 13</a></i>;<br/>
<a NAME="E3:22_10_3"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">D</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E4:22_10_3"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o'</font> where <font color="Olive">o'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> </span>}</span>  </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_10_3_1"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_10_3_1"/><i><font color="Green">E206</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:22_10_3_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>
<a NAME="E6:22_10_3_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o'</font> where <font color="Olive">o'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E7:22_10_3_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o'</font> where <font color="Olive">o'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> </span>}</span>  </span>}</span> 
 <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_10_3"/><b>then </b><i><font color="Green">E207</font></i>: 
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o'</font> where <font color="Olive">o'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> </span>}</span>  </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T46">LATTICE3:46</a></i>;<br/>
<a NAME="E6:22_10_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o'</font> where <font color="Olive">o'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> </span>}</span>  </span>}</span>  <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:22_10_3_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o'</font> where <font color="Olive">o'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> </span>}</span>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_10_3_2"/><i><font color="Green">E208</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o'</font> where <font color="Olive">o'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> </span>}</span>  )
 ;<br/>
<b>consider </b><font color="Maroon">ov</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E5:22_10_3_2"/><i><font color="Green">E210</font></i>: 
( <font color="Maroon">ov</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ov</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a></i>;<br/>
<b>thus </b><a NAME="E6:22_10_3_2"/>
<font color="Maroon">u</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="conlat_2.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_10_3"/><b>then </b>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o'</font> where <font color="Olive">o'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> </span>}</span>  </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E8:22_10_3"/>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o'</font> where <font color="Olive">o'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> </span>}</span>  </span>}</span> ,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:22_10"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:22"/><i><font color="Green">E211</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>;<br/>

<b>set </b><font color="Maroon">y</font> =  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font>;<br/>
<a NAME="E1:22_11"/><i><font color="Green">E212</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span>,<span class="p2">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 
;<br/>
<i><font color="Green">E213</font></i>: <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>
<br/>.= 
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>
;<br/>
<a NAME="E3:22_11"/><i><font color="Green">E214</font></i>: 
for <font color="Olive">o'</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/>for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o'</font> be   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:22_11_2"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">o'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">a'</font> be   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E2:22_11_2"/>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
 ;<br/>

<a NAME="E3:22_11_2"/><b>then </b>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1">ObjectDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D13">CONLAT_1:def 13</a></i>;<br/>
<a NAME="E4:22_11_2"/><b>then </b>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D6">CONLAT_1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">aa</font> being   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E6:22_11_2"/><i><font color="Green">E216</font></i>: 
( <font color="Maroon">aa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; ( for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">aa</font> ) )
 ;<br/>
<a NAME="E7:22_11_2"/>
<font color="Maroon">o'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">a'</font>
 
<b>by </b><i>, <a class="ref" href="conlat_2.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E8:22_11_2"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a'</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:22_11"/><i><font color="Green">E217</font></i>: 
for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> :  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span>  holds <br/><font color="Olive">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_11_3"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> :  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_11_3"/><i><font color="Green">E218</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> &amp;  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> )
 ;<br/>
<a NAME="E4:22_11_3"/><i><font color="Green">E219</font></i>: 
for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:22_11_3_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
 ;<br/>

<a NAME="E2:22_11_3_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o'</font></span>)</span> where <font color="Olive">o'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> 
 
;<br/>
<a NAME="E3:22_11_3_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T38">LATTICE3:38</a></i>;<br/>
<b>hence </b><a NAME="E4:22_11_3_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_11_3"/>
for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:22_11_3_2"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
 ;<br/>

<a NAME="E2:22_11_3_2"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E3:22_11_3_2"/>
<font color="Maroon">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">u</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_11_3"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a'</font> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o'</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Olive">o'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a'</font> </span>}</span> 
 
;<br/>
<a NAME="E7:22_11_3"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1">ObjectDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D6">CONLAT_1:def 6</a></i>;<br/>
<a NAME="E8:22_11_3"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D13">CONLAT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E9:22_11_3"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> 
 <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_11"/><i><font color="Green">E220</font></i>: 
for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span>  holds <br/><font color="Olive">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> :  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_11_4"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_11_4"/><i><font color="Green">E221</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> )
 ;<br/>
<a NAME="E4:22_11_4"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o'</font></span>)</span> where <font color="Olive">o'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span>  <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:22_11_4_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o'</font></span>)</span> where <font color="Olive">o'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_11_4_1"/><i><font color="Green">E222</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> )
 ;<br/>
<b>thus </b><a NAME="E4:22_11_4_1"/>
<font color="Maroon">q</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_11_4"/><b>then </b>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E6:22_11_4"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> :  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_11"/>
 <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> :  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> ,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_5"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">D</font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">d</font></span>)</span><b> such that </b><br/><a NAME="E2:22_11_5"/><i><font color="Green">E223</font></i>: 
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">D</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="lattice6.html#D14">LATTICE6:def 14</a></i>;<br/>
<a NAME="E3:22_11_5"/><i><font color="Green">E224</font></i>: 
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<a NAME="E4:22_11_5"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> :  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_11_5_1"/><i><font color="Green">E225</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_11_5_1"/><i><font color="Green">E226</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">d</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/>
<a NAME="E5:22_11_5_1"/>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 
<b>by </b><i>, , , <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<b>hence </b><a NAME="E6:22_11_5_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> :  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_11_5"/><b>then </b><i><font color="Green">E227</font></i>: 
 <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> :  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="lattice3.html#T46">LATTICE3:46</a></i>;<br/>
<a NAME="E6:22_11_5"/>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattice3.html#R3">is_less_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> :  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:22_11_5_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> :  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_11_5_2"/><i><font color="Green">E228</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> &amp;  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_11_5_2"/>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">u</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_11_5"/><b>then </b>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> :  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T40">LATTICE3:40</a></i>;<br/>
<b>hence </b><a NAME="E8:22_11_5"/>
 <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> :  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> ,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font></span>)</span> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:22_11"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i>, , <a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:22"/><i><font color="Green">E229</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> holds  <font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>;<br/>

<a NAME="E1:22_12"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">x</font> =  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">O</font> = <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> </span>}</span> ;<br/>
<a NAME="E2:22_12"/>
<span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_12_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">u'</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_12_1"/><i><font color="Green">E231</font></i>: 
( <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_12_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">O</font> = <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font> ;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">a'</font> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> :  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> ;<br/>
<a NAME="E4:22_12"/>
<span class="p1">{<span class="default"> <font color="Olive">a'</font> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> :  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_12_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a'</font> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> :  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">u'</font> being   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_12_2"/><i><font color="Green">E232</font></i>: 
( <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp;  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_12_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">a'</font> where <font color="Olive">a'</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> :  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> ;<br/>
<a NAME="E6:22_12"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font></span>)</span>,<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fi</font>
 
;<br/>
<a NAME="E7:22_12"/><b>then </b><i><font color="Green">E233</font></i>: 
<font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
 
<b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E8:22_12"/><b>then </b><i><font color="Green">E234</font></i>: 
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="conlat_1.html#T35">CONLAT_1:35</a></i>;<br/>
<a NAME="E9:22_12"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:22_12_3"/>
<font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_12_3_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 ;<br/>

<b>then consider </b><font color="Maroon">o'</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_12_3_1"/><i><font color="Green">E236</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o'</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> )
 ;<br/>
<a NAME="E4:22_12_3_1"/>
for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a'</font> be   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:22_12_3_1_1"/>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
 ;<br/>

<a NAME="E2:22_12_3_1_1"/><b>then </b>
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">y</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> 
 
;<br/>
<a NAME="E3:22_12_3_1_1"/><b>then </b>
 <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T38">LATTICE3:38</a></i>;<br/>
<b>hence </b><a NAME="E4:22_12_3_1_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a'</font>
 <b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:22_12_3_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:22_12_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">o'</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:22_12_3"/><i><font color="Green">E237</font></i>: 
( <font color="Maroon">o'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; ( for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> ) )
 ;<br/>
<a NAME="E5:22_12_3"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o'</font> <a href="lattice3.html#R3">is_less_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">y</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:22_12_3_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">y</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">qa</font> being   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_12_3_2"/><i><font color="Green">E239</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">qa</font> &amp; <font color="Maroon">qa</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> )
 ;<br/>
<b>thus </b><a NAME="E4:22_12_3_2"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o'</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_12_3"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o'</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<b>hence </b><a NAME="E7:22_12_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 <b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:22_12"/>
<span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:22_12_4"/>
<span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a'</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_12_4_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">u'</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:22_12_4_1"/><i><font color="Green">E240</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> &amp; ( for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> ) )
 ;<br/>
<a NAME="E4:22_12_4_1"/>
for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Maroon">u'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a'</font> be   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:22_12_4_1_1"/>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
 ;<br/>

<a NAME="E2:22_12_4_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a'</font>
 
<b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E3:22_12_4_1_1"/>
<font color="Maroon">u'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">a'</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:22_12_4_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a'</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:22_12_4"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">u'</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:22_12_4"/><i><font color="Green">E241</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> &amp; ( for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Maroon">u'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a'</font> ) )
 ;<br/>
<a NAME="E5:22_12_4"/>
for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a'</font> be   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:22_12_4_2"/>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
 ;<br/>

<a NAME="E2:22_12_4_2"/><b>then </b>
<font color="Maroon">u'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">a'</font>
 
<b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E3:22_12_4_2"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a'</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:22_12_4"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a'</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E242</font></i>: <font color="Maroon">O</font> = 
<span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="conlat_1.html#D7">CONLAT_1:def 7</a></i>
<br/>.= 
the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="conlat_1.html#D13">CONLAT_1:def 13</a></i>
;<br/>
<b>then </b><font color="Maroon">A</font> = 
<span class="p1">(<span class="default"><a href="conlat_1.html#K1">ObjectDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="conlat_1.html#D13">CONLAT_1:def 13</a></i>
<br/>.= 
the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="conlat_1.html#D13">CONLAT_1:def 13</a></i>
;<br/>
<b>hence </b><a NAME="E13:22_12"/>
<font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:22"/><i><font color="Green">E243</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>  st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font> holds <br/><font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:22_13"/><i><font color="Green">E244</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:22_13"/>
( <font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:22_13"/>
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:22"/><b>then </b><i><font color="Green">E245</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i/>;<br/>
<a NAME="E24:22"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_14"/><i><font color="Green">E246</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>

<a NAME="E2:22_14"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">fi</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:22_14"/><i><font color="Green">E247</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fi</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<a NAME="E5:22_14"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fi</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fi</font> )
 
<b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<i><font color="Green">E248</font></i>: <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>
<br/>.= 
<font color="Maroon">u</font>
<b>by </b><i/>
;<br/>
<a NAME="E9:22_14"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:22_14"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:22"/><b>then </b><i><font color="Green">E249</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="lattice4.html#M1">Homomorphism</a> of  <a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font>,<font color="Maroon">L</font> <b>by </b><i>, , , </i>;<br/>
<a NAME="E27:22"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">f</font> is <a href="lattice4.html#V2">epimorphism</a>
 
<b>by </b><i>, <a class="ref" href="lattice4.html#D3">LATTICE4:def 3</a></i>;<br/>
<a NAME="E28:22"/>
<font color="Maroon">f</font> is <a href="lattice4.html#V1">monomorphism</a>
 
<b>by </b><i>, <a class="ref" href="lattice4.html#D2">LATTICE4:def 2</a></i>;<br/>
<a NAME="E29:22"/><b>then </b>
<font color="Maroon">f</font> is <a href="lattice4.html#V3">isomorphism</a>
 
<b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="lattice4.html#D4">LATTICE4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E30:22"/>
 <a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font>,<font color="Maroon">L</font> <a href="filter_1.html#R1">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="lattice4.html#D5">LATTICE4:def 5</a></i>;<br/>


</div>
